FEBE1004A – EAD 1B Lecture Packages - Package #3 PDF

Faculty of Engineering and the Built Environment University of the Witwatersrand, Johannesburg FEBE1004A – EAD 1B Lecture Packages – Package #3 Data Analysis Topics 1. Types of graphs and their uses...

Faculty of Engineering and the Built Environment University of the Witwatersrand, Johannesburg FEBE1004A – EAD 1B Lecture Packages – Package #3 Data Analysis Topics 1. Types of graphs and their uses covered: 2. Straight and weighted averages 3. Histograms and their uses 4. Trend-lines 5. Significant figures 1. Types of Graphs and Their Uses Detailed Pie Chart - to represent fractions, ratios, percentages, etc. of a whole; for example, the constituents of a concrete mix, where each description: slice represents a contributing volume / mass to the total volume / mass. Bar graph - to show comparative results of several samples, for example, the strengths of several concrete test specimens. The sample identifiers are on the X-axis and the Y-axis is used to indicate the strengths. Scatter Plot - for showing the relationship of a dependent variable versus a controlled variable; for example, the strength of concrete vs time. The dependent variable (strength) would be on the Y-axis whilst the controlled / independent variable (time) would be on the X-axis. Page 1 of 6 Faculty of Engineering and the Built Environment University of the Witwatersrand, Johannesburg 1. Types of Graphs and Their Uses A mathematical model of such a relationship (or an infinite number of measured data points) would result in a solid / continuous line. However, a series of measured data points would result in a scatter of dots plotted against the X-axis. Data markers (such as: + x o *) are used to indicate where actual measurements have been taken, and when plotted together with a modelled relationship, provide confidence in the model. A trend line could also be used, based on the actual data points, to form a representative mathematical model. Several functions could be plotted against a common controlled variable (X-axis) such as the curing strengths of several concrete specimens; each specimen being represented with a different colour, or different style, of line / date point. Errors bars are used to indicate the range within which a data point could exist, based on the accuracy of the actual measurement. For example, if a weighing scale that could measure accurate to 10g was used to weigh a specimen of concrete and returned a reading of 200g, then in reality, the specimen could weigh anything between 195g and 205g. A pair of error bars showing these limits (typically looking like this: I ) is overlaid on the data point. The narrower the range of error, the better the accuracy of the data point; this provides more confidence in the measurements. The primary source of measurement error is the accuracy to which the instrument used, has been designed. In the old days, the needles of analogue instruments were hard to see, and this resulted in a reading error. But, due to today's digital readouts, this form of measurement error rarely exists. Page 2 of 6 Faculty of Engineering and the Built Environment University of the Witwatersrand, Johannesburg 2. Straight and Weighted Averages Detailed A straight average is used to represent a mean value where all samples have contributed equally. For example, the class average for description: a particular course is the mean value determined by the sum of each students' mark, divided by the number students in the class. A weighted average represents a mean value where some samples have more importance than others, and therefore contribute more significantly towards the final value. For example, your end-of-year mark for first year is more dependent on your PHYS1014 mark than that of your chosen elective, because PHYS1014 is a higher-credit course and is therefore weighted more in terms of importance. Page 3 of 6 Faculty of Engineering and the Built Environment University of the Witwatersrand, Johannesburg 3. Histograms and Their Uses Detailed This is a special type of bar graph that shows how many samples, of the total, fall within particular ranges (called bins). For example, description: the marks for a course might reveal that no students scored in the 0-20 range, none in the 20-40 range, five students scored in the 40-60 range, 15 students were in the 60-80 range and seven students were in the 80-100 range. In this example, five equally-sized bins have been used, and the total sum of students in each of the bins equals the class size i.e. 27 students. This provides far more information than the simple straight average. The shape of the histogram is often of more importance than the actual numbers themselves. For example, the classical bell curve of subject marks is an example of this. Histograms are often used in sampling and measurements, for example, measuring the wind speed at a particular location over time, for the development of a wind-farm. Design engineers would want to know, on average, for how much time does the wind blow over a certain strength, so that they can determine the size of the wind turbines required. For example in Johannesburg, for most of the time, the wind does not blow, but when it does, it is gale-force! This would be clearly evident using a histogram, but this evidence would be lost if a straight average was merely used. A weighted average could be used to better represent this data by giving higher wind-speeds more importance. The accuracy of a histogram depends on the number of bins used, which is inversely related to the size of each bin i.e. many smaller bins is more accurate than fewer larger bins. The shape of the histogram is often represented by a distribution curve such as normal, Gaussian, natural, etc. and is usually of primary concern in engineering. Page 4 of 6 Faculty of Engineering and the Built Environment University of the Witwatersrand, Johannesburg 4. Trend-Lines Detailed These are a useful tool to make forecasts or predictions based on a batch of measured data points. For example, recordings of description: population sizes over the last 10 years would probably show a linear or exponential growth in the future, if a trend-line was determined based on the past. Another very useful aspect of inserting a trend-line into measured data, is the ability to determine a representative mathematical function, that best suites the data. Once an equation has been determined, this mathematical model offers major benefits to engineering. The ability to best-fit any data is represented by an R2 value; the closer this value is to 1, the better the trend-line represents the data, once again providing confidence. In Excel for example, when inserting a trend-line, one has the ability to select a linear (straight line), power, exponential, logarithmic, etc. function that provides a best-fit. Experiment with a simple example in Excel. Page 5 of 6 Faculty of Engineering and the Built Environment University of the Witwatersrand, Johannesburg 5. Significant Figures Detailed In general, the order-of-magnitude has more significance in a calculation than the actual value itself. Numbers written in scientific description: notation are easier to compare, in terms of significance, than when expressed as decimals. For example, when comparing R1 000 000 (1x106) to R20 (2x101), the fact that the first number has five more zeroes than the second number, is what primarily determines the significant difference between the two. Adding them together makes no difference to the answer, as whether one wins R1 000 000 in the Lotto or R1 000 020 both amounts make one a millionaire - the R20 would probably not even be considered. However, R20 (2x101) compared to 2c (2x10-2) now has much more significance! When used in conjunction with a logarithm, this forms the basis of a decibel: an important comparative technique commonly used in measurements. Page 6 of 6